Question

how would you evaluate $int\tan^{20}xsec^{2}x dx$?

choose the correct answer.

a. use u = tan x.
b. rewrite the power of tan x in terms of sec x. then use u = sec²x.
c. use u = sec x.
d. rewrite the power of sec x in terms of tan x. then use u = tan²⁰x.

Explanation:

Step1: Recall derivative of tangent

The derivative of $\tan x$ is $\sec^{2}x$, i.e., $d(\tan x)=\sec^{2}x dx$.

Step2: Apply substitution

In the integral $\int\tan^{20}x\sec^{2}x dx$, if we let $u = \tan x$, then $du=\sec^{2}x dx$. The integral becomes $\int u^{20}du$.

Answer:

A. Use $u = \tan x$